For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
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2answers
152 views

Given two basis, find the transformation matrix from one to another

I have these two basis of $M^R_{2x2}$: $C= (\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 \\ 1 & 0\end{...
3
votes
1answer
69 views

Two Identities Involving Trace

Let $x$ by a $p \times 1$ vector and $M$, $Q$ be two $p\times p$ matrix. Then it is claimed that $$ \mbox{trace }\left( M^{-1}xx^TM^{-1} \right) = \left\| M^{-1}x \right\|^2, $$ and $$ \mbox{trace }\...
2
votes
3answers
127 views

If $AB\not=BA,$ then do $A$ and $B$ have to be matrices

Is this solvable? Or are there other things that fit the bill for $A$ and $B$?
4
votes
1answer
210 views

What exacty is the role played by Jacobian or Wronskian?

In many of our derivations or in differential equations we come across the terms Jacobian or Wronskian. For example, to check the linear independence of solutions of differential equations, we ensure ...
0
votes
1answer
70 views

the examples of subspace embedding which are not Oblivious

For the definitions of Oblivious Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf. Then, can any one show the examples of subspace embedding which are ...
0
votes
2answers
29 views

If I want to find the dimension of the image of a linear transformation…

If I have a linear transformation $T(v)=Av$ and want to find the dimension of the range$(T)$, the following procedure is valid? Looking at the columns of $A$, if all columns are linearly independent, ...
2
votes
0answers
53 views

Eigenvalue multiplicity of a product of two real skew-symmetric matrices

All the roots of characteristics polynomial of $AB$, where $A$, $B$ are skew symmetric matrices of order $2n$, are of multiplicity greater then $1$. I know that eigenvalues of skew symmetric matrices ...
-2
votes
1answer
46 views

Traceless nilpotent matrices

A traceless, nilpotent $2 \times 2$ matrix with one real variable is already known. Does anyone know of a $3 \times 3$ one, containing at least one real variable or parameter? A nice set of ...
1
vote
2answers
59 views

Canonical matrix of a transformation

Let $T: \mathbb{R}^2 \to \mathbb{R}^3$ and $T(-2,3)=(-1,0,1)$ and $T(1,2)=(0,-1,0)$. Obtain the canonical matrix of $T$ and the transformation $T(x,y)$. How I obtain the canonical matrix of a ...
0
votes
2answers
46 views

Show that $G = SL_3(F_p)$ acts transitively on $H = F_p^3\setminus \lbrace0\rbrace$

I need to show that $G = SL_3(F_p)$ acts transitively on $H = F_p^3\setminus \lbrace0\rbrace$. That means to show that, for all $s,t \in H$, there is $g \in G$ such that $gt = s$. I tried to make ...
2
votes
0answers
78 views

The Maximum Singular Value of a Certain $\{1,-1\}$-Matrix

I have a $16$-dimensional real symmetric matrix with entries in $\{1,-1\}$. $11$ of the rows are pairwise orthogonal, so are the remaining $5$ rows. But the two orthogonal sets are not necessarily ...
2
votes
1answer
28 views

different ways to see why this matrix limit is correct

given that $0 < a < 1$ it follows that: $$\lim_{n\to\infty}\begin{pmatrix} a & (1-a) \\ (1-a) & a \end{pmatrix}^n = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix}$$...
0
votes
0answers
58 views

Efficient method to compute grand sum of a Vandermonde matrix?

Is there a computationally efficient method to calculate the sum of all elements (grand sum) of a Vandermonde matrix? Each row can be quickly calculated using the formula for a geometric progression. ...
2
votes
0answers
62 views

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$. How to prove this result? (I found this statement while ...
1
vote
1answer
88 views

Cycles in the Fibonacci Sequence mod n with matrices

I was just looking at this question about Fibonacci sequence cycles modulo 5, and I happened to see a very nice solution that involved using matrices. Using the matrix representation of the Fibonacci ...
2
votes
0answers
41 views

Matrix for which $|a_{ii}| \leq \sum_{j\neq i} |a_{ij}|$ for all i.

From Wikipedia, a matrix is said to be diagonally dominant if for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the ...
2
votes
0answers
74 views

Signs of real part eigenvalues for nonsymmetric matrix.

Since the search of the eigenvalues is in general not "simple", equally valid, is the method of reducing with moves of Gauss, that preserve the determinant, (add to multiple rows of other rows, move ...
3
votes
2answers
39 views

An “apparent” contradiction for eigenvalues signs of $A=\left( \begin{array}{cc} a & -a \\ a-1 & 1-a \\ \end{array} \right)$.

The following matrix $$A=\left( \begin{array}{cc} a & -a \\ a-1 & 1-a \\ \end{array} \right)$$ has eigenvalues $\lambda=0,1$ $\forall a\in\mathbb R$. Therefore $\lambda\geq 0$. ...
0
votes
1answer
53 views

Inequality about Frobenius norms for matrices [closed]

For any square matrix A, but not necessarily symmetric, what are some ways to prove the inequality $$ \|A^2\|_F^2\leq\|A^TA\|_F^2, $$ where $\|B\|_F^2=tr(B^TB)$ is the Frobenius norm of matrix $B$ ?
2
votes
1answer
57 views

Matrix and Abelian groups question

Let $A$ be a Matrix: $$ A=\begin{pmatrix} 1 & 2\\ 4 & 1 \end{pmatrix} $$ Let $f\colon v\to Av$ be a homomorphism from $Z^2$ to $Z^2$. Find a base $(v_1,v_2)$ to $Z^2$ and $2$ integers $...
0
votes
1answer
44 views

Use well-known relation: $\det(I + \epsilon X) = 1 + \operatorname{tr}(X) \epsilon + O(\epsilon^2).$ for evaluate $\det(I + M)$.

Let $d_1$, $d_2\in[0,1]$. Use well-known relation: $$\det(I + \epsilon X) = 1 + \operatorname{tr}(X) \epsilon + O(\epsilon^2).$$ for evaluate $$\det(I + M)$$ where $I$ is the $2\times2$ identity ...
0
votes
1answer
72 views

invertibility of $A^{-1} + B^{-1}$ [duplicate]

Let $A$ and $B$ be two invertible $n\times n$ real matrices. Assume that $A+B$ is invertible. Show that $A^{-1} + B^{-1}$ is also invertible.
0
votes
1answer
65 views

If columns/rows of an $n\times n$ matrix $M$ are linearly independent what is the rank of $M$?

1.) I think I can answer the case when the rows are linearly independent vectors: Since the rows of the matrix $M$ are linearly independent, we cannot create an all $0$ row in the matrix therefore ...
0
votes
0answers
26 views

One of the following two matrix is symmetric positive definite?

Recently, I com across a questions, which can be given as follows, If there are two symmetric positive semi-definite matrix $W$ and $T$, but they satisfy the following condition: $null(W)\cap null(T)...
0
votes
1answer
47 views

Matrix Transformation

Consider a triangle that has vertices at $A(1, 2)$, $B(2, 3)$ and $C(4, 2)$. Reflect this triangle in the line through the origin which is inclined at $30^\circ$ to the positive $x$-axis. Find the ...
4
votes
1answer
105 views

Show that trace is a unique linear functional [duplicate]

If $W$ is the space of $n \times n$ matrices over a field, and $f$ is a linear functional on $W$ such that $f(AB)=f(BA)$ for each $A,B$ in $W$, and $f(I)=n$, then $f$ is the trace function. I have ...
1
vote
1answer
36 views

Suppose $A$ is an $n \times n$ matrix. Prove that similarity transformation is reflexive. ie $A=P^{-1}AP $

I know a correct answer to this question is to just assume that $P$ is the identity matrix. But, is this argument BELOW correct. $$A=A \Rightarrow AP=AP\Rightarrow A=P^{-1}AP $$
0
votes
1answer
61 views

Computation of transformation matrix for jordan normal form: how to choose eigenvectors

During this semester at university we we're introduced to the jordan normal form of a matrix. While we never wrote down an explicit algorithm of how to find the matrix $B$, such that $B^{-1}AB$ is a ...
0
votes
1answer
43 views

What is called the following (matrix) operator?

Let $\mathbf{A}=\begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1m} \\ A_{21} & A_{22} & \cdots & A_{2m} \\ \vdots & \vdots & \cdots & \vdots \\ A_{m1} & A_{m2} &...
0
votes
0answers
25 views

Impose linear constraints

I am implementing Han and Kanade's [1] perspective factorization method. I reached the point where I need to impose some constraints on the Motion matrix so it becomes valid. The matrix I want to ...
0
votes
1answer
498 views

All principal minors are equal to zero

Let's assume that all principal minors of symetric square matrix $A$ ($n\times n$) are equal to zero, then what definiteness does this matrix have? It's obvious that it's semidefinite, of course. But ...
0
votes
0answers
54 views

Multivariable polynomial matrix representations

This is a follow-up to matrix representation of parabola and matrix representation to generate monomials. I found a method to build such matrices to implement this type of functionality for one ...
1
vote
1answer
69 views

Calculating a stochastic matrix with multiple states

I am struggling with how to calculate the values of a Markov matrix which has multiple states. For example, Imagine an unfair 6 sided dice. The chance of rolling a 1,2,3,4,5 or 6 is 0.3, 0.25, 0.2, ...
2
votes
3answers
185 views

$A$ is a real orthogonal matrix, prove that $ (I+A)^{-1}(I-A)$ is skew-symmetric

$A$ is a real orthogonal matrix and $(I + A)$ is non-singular. Prove that $ (I+A)^{-1}(I-A)$ is a skew-symmetric matrix. Attempt: $[(I+A)^{-1}(I-A)]^t=(I-A)^t((I+A)^{-1})^t=(I-A)^t((I+A)^{t})^{-1}=(...
3
votes
1answer
32 views

$n$ dimensional determinant using recurrence relations

Find determinant $$D_n(a,b,c)= \begin{vmatrix} a & b & 0 & 0 & \cdots & 0 & 0 & 0 \\ c & a & b & 0 & \cdots & 0 & 0 & 0 ...
3
votes
1answer
118 views

Why is the determinant of $\sum_{i=1}^n A_i^2$ non-negative?

Let $A_i$ be an $n\times n$ matrix in $\mathbb{R}$ and $\{A_i\}_{i=1}^k$ are pairwise commutative: $A_iA_j = A_jA_i$. How to show $det(\sum_{i=1}^k A_i^2)\geq 0$? We may consider this question in ...
1
vote
1answer
28 views

Dimension of $Range(A)$ and $Range(A^2)$

Which of the following matrices satisfies the property "Dimension of $Range(A)$ and $Range(A^2)$ is 2 and 1 respectively." $$A = \begin{bmatrix} 0 & 0 & 1 & 1\\ 0 & 0 & 1 & 1\...
0
votes
0answers
58 views

multiplication between a matrix and a givens rotation

I want to multiply a matrix A with a givens rotation G. As a reference to this very important link:Click here, they explained in pages 13 and 14 how this multiplication can be achieved. In this PDF, ...
2
votes
0answers
287 views

Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as $$\text{...
0
votes
1answer
42 views

Graph Theory : Strongly regular graph

A simple graph G which is neither empty nor complete is said to be strongly regular with parameters $(v,k,λ,μ)$ if: v(G)=v; G is k-regular; any two adjacent vertices of G have λ common neighbours, ...
0
votes
1answer
103 views

Bordered minor and rank of a matrix

Let $M\in\mathbf{R}^{n\times n}$ be a matrix. Suppose that there is a $k\times k$ minor $M_k$ of rank k. Now this reference (Algebra For Iit Jee 7.65) here states that if all the $k+1$th minors ...
0
votes
3answers
59 views

Getting $x,y$ position on an image based on given value

This should be simple but my math skills are really bad ... I have an image of 36 images (6 by 6 matrix). These small images are 36 instances of a direction arrow (like from Google maps GPS), each ...
-1
votes
1answer
58 views

Super Simple Proof of Cofactor Expansion

Is it possible to provide a super simple proof that cofactor expansion gives a determinant value no matter which row or column of the matrix you expand upon? E.g., super simply prove that $$\det(A) =...
0
votes
1answer
20 views

Is there a matrix product which results in this relation?

Let $\pmb{a} = \left[a_1\;a_2\;\dots\;a_n\right], \pmb{b} = \left[b_1\;b_2\;\dots\;b_m\right]$. Then $$K = \left( \begin{array}{ccc} a_1b_1 & a_1b_2 & \cdots & a_1b_m \\ a_2b_1 & ...
0
votes
1answer
251 views

Block Gauss -Seidel Iterative Method for Overdetermined Linear Systems

I am interested in solving a large linear system with block Gauss-Seidel Iterative Method. Suppose I have the following block matrix $A$ for $Ax=b$ linear system: \begin{bmatrix} B & 0 & C &...
2
votes
0answers
82 views

Fastest way to find linearly independent columns of a matrix

Given a rectangular matrix $X$ of size $n\times m$ with $m>n$, what is the fastest way to find the linearly independent coloums. Robust methods like SVD or RRQR decompostion have complexity of ...
-3
votes
1answer
46 views

Entries of a matrix?

The wikipedia article on matrices states: "..is a rectangular array—of numbers, symbols, or expressions,.." https://en.wikipedia.org/wiki/Matrix_(mathematics) Do you agree with that? Are the ...
3
votes
3answers
469 views

Find all constants where a matrix is symmetric

I have a matrix like below: $$M = \begin{bmatrix} 2 && a-2b+c && 2a+b+c \\ 3 && 5 && -2 \\ 0 && a+c && 7\end{bmatrix}$$ In order for the matrix to be ...
1
vote
1answer
54 views

Function composition as representable by matrices?

I know from linear algebra that for different sets of functions differentiation can be expressed using matrix multiplication on a vector representation of the function. For instance polynomials and ...
1
vote
1answer
88 views

Prove that $\det(E^{\mathsf{T}})=\det(E)$, where $E$ is an $n \times n$ elementary matrix.

Prove that $\det(E^{\mathsf{T}})=\det(E)$, where $E$ is an $n \times n$ elementary matrix. I am not familiar with eigenvalues. I only know that the cofactor expansion can be done along any row of a ...